Consider the solution $\phi(x, y)$ of Laplace's equation in two dimensions using a relaxation method on a square grid with common spacing $h$. As in the main text, denote $\phi\left(x_{0}+i h, y_{0}+j h\right)$ by $\phi_{i, j} .$ Further, define $\phi_{i, j}^{m, n}$ by
$$
\phi_{i, j}^{m, n} \equiv \frac{\partial^{m+n} \phi}{\partial x^{m} \partial y^{n}}
$$
evaluated at $\left(x_{0}+i h, y_{0}+j h\right)$.
(a) Show that
$$
\phi_{i, j}^{4,0}+2 \phi_{i, j}^{2,2}+\phi_{i, j}^{0,4}=0
$$
(b) Working up to terms of order $h^{5}$, find Taylor series expansions, expressed in terms of the $\phi_{i, j}^{m, n}$, for
$$
\begin{aligned}
&S_{\pm, 0}=\phi_{i+1, j}+\phi_{i-1, j} \\
&S_{0, \pm}=\phi_{i, j+1}+\phi_{i, j-1}
\end{aligned}
$$
(c) Find a corresponding expansion, to the same order of accuracy, for $\phi_{i \pm 1, j+1}+$ $\phi_{i \pm 1, j-1}$ and hence show that
$$
S_{\pm, \pm}=\phi_{i+1, j+1}+\phi_{i+1, j-1}+\phi_{i-1, j+1}+\phi_{i-1, j-1}
$$
has the form
$$
4 \phi_{i, j}^{0,0}+2 h^{2}\left(\phi_{i, j}^{2,0}+\phi_{i, j}^{0,2}\right)+\frac{h^{4}}{6}\left(\phi_{i, j}^{4,0}+6 \phi_{i, j}^{2,2}+\phi_{i, j}^{0,4}\right).
$$
(d) Evaluate the expression $4\left(S_{\pm, 0}+S_{0, \pm}\right)+S_{\pm, \pm}$and hence deduce that a possible relaxation scheme, good to the fifth order in $h$, is to recalculate each $\phi_{i, j}$ as the weighted mean of the current values of its four nearest neighbours (each with weight $\frac{1}{5}$ ) and its four next-nearest neighbours (each with weight $\frac{1}{20}$ ).